symmetric monoidal (∞,1)-category of spectra
A quasi-triangular bialgebra / triangular bialgebra is a bialgebra equipped with just the right structure such as to make its category of modules into a braided monoidal category/symmetric monoidal category.
Let $A$ be an algebra in a symmetric monoidal category $C$ with symmetry $\tau$; fix $m,l$ and $D\in A^{\otimes k}$ and let $1\leq i_r\leq l$ for $1\leq r\leq m$ be different. Then denote $D_{i_1,\ldots,i_m}\in A^{\otimes n}$ as the image of $R\otimes 1^{\otimes (l-m)}$ under the permutation which is the composition of the $m$ transpositions $(r,i_r)$ of tensor factors interchanging $r$ and $i_r$. In the following $C$ is the monoidal category of $k$-vector spaces.
A $k$-bialgebra (in particular $k$-Hopf algebra) is quasitriangular if there is an invertible element $R\in H\otimes H$ such that for any $h\in H$
where $\tau=\tau_{H,H}:H\otimes H\to H\otimes H$ and
An invertible element $R$ satisfying these 3 properties is called the universal $R$-element. As a corollary
and the quantum Yang-Baxter equation holds in the form
A quasitriangular $H$ is called triangular if $R_{21}:=\tau(R) = R^{-1}$.
The category of representations of a quasitrianguar bialgebra is a braided monoidal category. If $R$ is a universal $R$-element, then $R_{21}^{-1}$ is as well. If $H$ is quasitriangular, $H^{cop}$ and $H_{op}$ are as well, with the universal $R$-element being $R_{21}$, or instead, $R_{12}^{-1}$. Any twisting of a quasitriangular bialgebra by a bialgebra 2-cocycle twists the universal $R$-element as well; hence the twisted bialgebra is again quasitriangular. Often the $R$-element does not exist as an element in $H\otimes H$ but rather in some completion of the tensor square; we say that $H$ is essentially quasitriangular, this is true for quantized enveloping algebras $U_q(G)$ in the rational form. The famous Sweedler’s Hopf algebra has a 1-parameter family of universal $R$-matrices.
A quasitriangular structure on a bialgebra corresponds to a braided monoidal category structure on the category of modules of the underlying algebra. (For instance chapter 1, section 2 of (Carroll)).
Tannaka duality for categories of modules over monoids/associative algebras
monoid/associative algebra | category of modules |
---|---|
$A$ | $Mod_A$ |
$R$-algebra | $Mod_R$-2-module |
sesquialgebra | 2-ring = monoidal presentable category with colimit-preserving tensor product |
bialgebra | strict 2-ring: monoidal category with fiber functor |
Hopf algebra | rigid monoidal category with fiber functor |
hopfish algebra (correct version) | rigid monoidal category (without fiber functor) |
weak Hopf algebra | fusion category with generalized fiber functor |
quasitriangular bialgebra | braided monoidal category with fiber functor |
triangular bialgebra | symmetric monoidal category with fiber functor |
quasitriangular Hopf algebra (quantum group) | rigid braided monoidal category with fiber functor |
triangular Hopf algebra | rigid symmetric monoidal category with fiber functor |
supercommutative Hopf algebra (supergroup) | rigid symmetric monoidal category with fiber functor and Schur smallness |
form Drinfeld double | form Drinfeld center |
trialgebra | Hopf monoidal category |
2-Tannaka duality for module categories over monoidal categories
monoidal category | 2-category of module categories |
---|---|
$A$ | $Mod_A$ |
$R$-2-algebra | $Mod_R$-3-module |
Hopf monoidal category | monoidal 2-category (with some duality and strictness structure) |
3-Tannaka duality for module 2-categories over monoidal 2-categories
monoidal 2-category | 3-category of module 2-categories |
---|---|
$A$ | $Mod_A$ |
$R$-3-algebra | $Mod_R$-4-module |
V. G. Drinfel’d, Quantum groups, Proc. ICM 1986, Vol. 1, 2 798–820, AMS 1987.
S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. j. mod. physics A, 5, 01, pp. 1-91 (1990) doi:10.1142/S0217751X90000027
S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.
A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.
V. Chari, A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994
Robert Carroll, Calculus revisited